--- a/doc-src/IsarAdvanced/Functions/Thy/document/Functions.tex Tue Mar 03 17:05:18 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1985 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{Functions}%
-%
-\isadelimtheory
-\isanewline
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Functions\isanewline
-\isakeyword{imports}\ Main\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isamarkupsection{Function Definitions for Dummies%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-In most cases, defining a recursive function is just as simple as other definitions:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{fun}\isamarkupfalse%
-\ fib\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}fib\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}fib\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}fib\ {\isacharparenleft}Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ fib\ n\ {\isacharplus}\ fib\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-The syntax is rather self-explanatory: We introduce a function by
- giving its name, its type,
- and a set of defining recursive equations.
- If we leave out the type, the most general type will be
- inferred, which can sometimes lead to surprises: Since both \isa{{\isadigit{1}}} and \isa{{\isacharplus}} are overloaded, we would end up
- with \isa{fib\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ {\isacharprime}a{\isacharcolon}{\isacharcolon}{\isacharbraceleft}one{\isacharcomma}plus{\isacharbraceright}}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The function always terminates, since its argument gets smaller in
- every recursive call.
- Since HOL is a logic of total functions, termination is a
- fundamental requirement to prevent inconsistencies\footnote{From the
- \qt{definition} \isa{f{\isacharparenleft}n{\isacharparenright}\ {\isacharequal}\ f{\isacharparenleft}n{\isacharparenright}\ {\isacharplus}\ {\isadigit{1}}} we could prove
- \isa{{\isadigit{0}}\ {\isacharequal}\ {\isadigit{1}}} by subtracting \isa{f{\isacharparenleft}n{\isacharparenright}} on both sides.}.
- Isabelle tries to prove termination automatically when a definition
- is made. In \S\ref{termination}, we will look at cases where this
- fails and see what to do then.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Pattern matching%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\label{patmatch}
- Like in functional programming, we can use pattern matching to
- define functions. At the moment we will only consider \emph{constructor
- patterns}, which only consist of datatype constructors and
- variables. Furthermore, patterns must be linear, i.e.\ all variables
- on the left hand side of an equation must be distinct. In
- \S\ref{genpats} we discuss more general pattern matching.
-
- If patterns overlap, the order of the equations is taken into
- account. The following function inserts a fixed element between any
- two elements of a list:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{fun}\isamarkupfalse%
-\ sep\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}sep\ a\ {\isacharparenleft}x{\isacharhash}y{\isacharhash}xs{\isacharparenright}\ {\isacharequal}\ x\ {\isacharhash}\ a\ {\isacharhash}\ sep\ a\ {\isacharparenleft}y\ {\isacharhash}\ xs{\isacharparenright}{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}sep\ a\ xs\ \ \ \ \ \ \ {\isacharequal}\ xs{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-Overlapping patterns are interpreted as \qt{increments} to what is
- already there: The second equation is only meant for the cases where
- the first one does not match. Consequently, Isabelle replaces it
- internally by the remaining cases, making the patterns disjoint:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{thm}\isamarkupfalse%
-\ sep{\isachardot}simps%
-\begin{isamarkuptext}%
-\begin{isabelle}%
-sep\ a\ {\isacharparenleft}x\ {\isacharhash}\ y\ {\isacharhash}\ xs{\isacharparenright}\ {\isacharequal}\ x\ {\isacharhash}\ a\ {\isacharhash}\ sep\ a\ {\isacharparenleft}y\ {\isacharhash}\ xs{\isacharparenright}\isasep\isanewline%
-sep\ a\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}\isasep\isanewline%
-sep\ a\ {\isacharbrackleft}v{\isacharbrackright}\ {\isacharequal}\ {\isacharbrackleft}v{\isacharbrackright}%
-\end{isabelle}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\noindent The equations from function definitions are automatically used in
- simplification:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ {\isachardoublequoteopen}sep\ {\isadigit{0}}\ {\isacharbrackleft}{\isadigit{1}}{\isacharcomma}\ {\isadigit{2}}{\isacharcomma}\ {\isadigit{3}}{\isacharbrackright}\ {\isacharequal}\ {\isacharbrackleft}{\isadigit{1}}{\isacharcomma}\ {\isadigit{0}}{\isacharcomma}\ {\isadigit{2}}{\isacharcomma}\ {\isadigit{0}}{\isacharcomma}\ {\isadigit{3}}{\isacharbrackright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsubsection{Induction%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Isabelle provides customized induction rules for recursive
- functions. These rules follow the recursive structure of the
- definition. Here is the rule \isa{sep{\isachardot}induct} arising from the
- above definition of \isa{sep}:
-
- \begin{isabelle}%
-{\isasymlbrakk}{\isasymAnd}a\ x\ y\ xs{\isachardot}\ {\isacharquery}P\ a\ {\isacharparenleft}y\ {\isacharhash}\ xs{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharquery}P\ a\ {\isacharparenleft}x\ {\isacharhash}\ y\ {\isacharhash}\ xs{\isacharparenright}{\isacharsemicolon}\ {\isasymAnd}a{\isachardot}\ {\isacharquery}P\ a\ {\isacharbrackleft}{\isacharbrackright}{\isacharsemicolon}\ {\isasymAnd}a\ v{\isachardot}\ {\isacharquery}P\ a\ {\isacharbrackleft}v{\isacharbrackright}{\isasymrbrakk}\isanewline
-{\isasymLongrightarrow}\ {\isacharquery}P\ {\isacharquery}a{\isadigit{0}}{\isachardot}{\isadigit{0}}\ {\isacharquery}a{\isadigit{1}}{\isachardot}{\isadigit{0}}%
-\end{isabelle}
-
- We have a step case for list with at least two elements, and two
- base cases for the zero- and the one-element list. Here is a simple
- proof about \isa{sep} and \isa{map}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ {\isachardoublequoteopen}map\ f\ {\isacharparenleft}sep\ x\ ys{\isacharparenright}\ {\isacharequal}\ sep\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharparenleft}map\ f\ ys{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}induct\ x\ ys\ rule{\isacharcolon}\ sep{\isachardot}induct{\isacharparenright}%
-\begin{isamarkuptxt}%
-We get three cases, like in the definition.
-
- \begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}a\ x\ y\ xs{\isachardot}\isanewline
-\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }map\ f\ {\isacharparenleft}sep\ a\ {\isacharparenleft}y\ {\isacharhash}\ xs{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ sep\ {\isacharparenleft}f\ a{\isacharparenright}\ {\isacharparenleft}map\ f\ {\isacharparenleft}y\ {\isacharhash}\ xs{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\isanewline
-\isaindent{\ {\isadigit{1}}{\isachardot}\ \ \ \ }map\ f\ {\isacharparenleft}sep\ a\ {\isacharparenleft}x\ {\isacharhash}\ y\ {\isacharhash}\ xs{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ sep\ {\isacharparenleft}f\ a{\isacharparenright}\ {\isacharparenleft}map\ f\ {\isacharparenleft}x\ {\isacharhash}\ y\ {\isacharhash}\ xs{\isacharparenright}{\isacharparenright}\isanewline
-\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}a{\isachardot}\ map\ f\ {\isacharparenleft}sep\ a\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\ sep\ {\isacharparenleft}f\ a{\isacharparenright}\ {\isacharparenleft}map\ f\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\isanewline
-\ {\isadigit{3}}{\isachardot}\ {\isasymAnd}a\ v{\isachardot}\ map\ f\ {\isacharparenleft}sep\ a\ {\isacharbrackleft}v{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\ sep\ {\isacharparenleft}f\ a{\isacharparenright}\ {\isacharparenleft}map\ f\ {\isacharbrackleft}v{\isacharbrackright}{\isacharparenright}%
-\end{isabelle}%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ auto\ \isanewline
-\isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-With the \cmd{fun} command, you can define about 80\% of the
- functions that occur in practice. The rest of this tutorial explains
- the remaining 20\%.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{fun vs.\ function%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The \cmd{fun} command provides a
- convenient shorthand notation for simple function definitions. In
- this mode, Isabelle tries to solve all the necessary proof obligations
- automatically. If any proof fails, the definition is
- rejected. This can either mean that the definition is indeed faulty,
- or that the default proof procedures are just not smart enough (or
- rather: not designed) to handle the definition.
-
- By expanding the abbreviation to the more verbose \cmd{function} command, these proof obligations become visible and can be analyzed or
- solved manually. The expansion from \cmd{fun} to \cmd{function} is as follows:
-
-\end{isamarkuptext}
-
-
-\[\left[\;\begin{minipage}{0.25\textwidth}\vspace{6pt}
-\cmd{fun} \isa{f\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}}\\%
-\cmd{where}\\%
-\hspace*{2ex}{\it equations}\\%
-\hspace*{2ex}\vdots\vspace*{6pt}
-\end{minipage}\right]
-\quad\equiv\quad
-\left[\;\begin{minipage}{0.48\textwidth}\vspace{6pt}
-\cmd{function} \isa{{\isacharparenleft}}\cmd{sequential}\isa{{\isacharparenright}\ f\ {\isacharcolon}{\isacharcolon}\ {\isasymtau}}\\%
-\cmd{where}\\%
-\hspace*{2ex}{\it equations}\\%
-\hspace*{2ex}\vdots\\%
-\cmd{by} \isa{pat{\isacharunderscore}completeness\ auto}\\%
-\cmd{termination by} \isa{lexicographic{\isacharunderscore}order}\vspace{6pt}
-\end{minipage}
-\right]\]
-
-\begin{isamarkuptext}
- \vspace*{1em}
- \noindent Some details have now become explicit:
-
- \begin{enumerate}
- \item The \cmd{sequential} option enables the preprocessing of
- pattern overlaps which we already saw. Without this option, the equations
- must already be disjoint and complete. The automatic completion only
- works with constructor patterns.
-
- \item A function definition produces a proof obligation which
- expresses completeness and compatibility of patterns (we talk about
- this later). The combination of the methods \isa{pat{\isacharunderscore}completeness} and
- \isa{auto} is used to solve this proof obligation.
-
- \item A termination proof follows the definition, started by the
- \cmd{termination} command. This will be explained in \S\ref{termination}.
- \end{enumerate}
- Whenever a \cmd{fun} command fails, it is usually a good idea to
- expand the syntax to the more verbose \cmd{function} form, to see
- what is actually going on.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{Termination%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\label{termination}
- The method \isa{lexicographic{\isacharunderscore}order} is the default method for
- termination proofs. It can prove termination of a
- certain class of functions by searching for a suitable lexicographic
- combination of size measures. Of course, not all functions have such
- a simple termination argument. For them, we can specify the termination
- relation manually.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{The {\tt relation} method%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Consider the following function, which sums up natural numbers up to
- \isa{N}, using a counter \isa{i}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ sum\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}sum\ i\ N\ {\isacharequal}\ {\isacharparenleft}if\ i\ {\isachargreater}\ N\ then\ {\isadigit{0}}\ else\ i\ {\isacharplus}\ sum\ {\isacharparenleft}Suc\ i{\isacharparenright}\ N{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ pat{\isacharunderscore}completeness\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\noindent The \isa{lexicographic{\isacharunderscore}order} method fails on this example, because none of the
- arguments decreases in the recursive call, with respect to the standard size ordering.
- To prove termination manually, we must provide a custom wellfounded relation.
-
- The termination argument for \isa{sum} is based on the fact that
- the \emph{difference} between \isa{i} and \isa{N} gets
- smaller in every step, and that the recursion stops when \isa{i}
- is greater than \isa{N}. Phrased differently, the expression
- \isa{N\ {\isacharplus}\ {\isadigit{1}}\ {\isacharminus}\ i} always decreases.
-
- We can use this expression as a measure function suitable to prove termination.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{termination}\isamarkupfalse%
-\ sum\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}relation\ {\isachardoublequoteopen}measure\ {\isacharparenleft}{\isasymlambda}{\isacharparenleft}i{\isacharcomma}N{\isacharparenright}{\isachardot}\ N\ {\isacharplus}\ {\isadigit{1}}\ {\isacharminus}\ i{\isacharparenright}{\isachardoublequoteclose}{\isacharparenright}%
-\begin{isamarkuptxt}%
-The \cmd{termination} command sets up the termination goal for the
- specified function \isa{sum}. If the function name is omitted, it
- implicitly refers to the last function definition.
-
- The \isa{relation} method takes a relation of
- type \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set}, where \isa{{\isacharprime}a} is the argument type of
- the function. If the function has multiple curried arguments, then
- these are packed together into a tuple, as it happened in the above
- example.
-
- The predefined function \isa{{\isachardoublequote}measure\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a{\isacharparenright}\ set{\isachardoublequote}} constructs a
- wellfounded relation from a mapping into the natural numbers (a
- \emph{measure function}).
-
- After the invocation of \isa{relation}, we must prove that (a)
- the relation we supplied is wellfounded, and (b) that the arguments
- of recursive calls indeed decrease with respect to the
- relation:
-
- \begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ wf\ {\isacharparenleft}measure\ {\isacharparenleft}{\isasymlambda}{\isacharparenleft}i{\isacharcomma}\ N{\isacharparenright}{\isachardot}\ N\ {\isacharplus}\ {\isadigit{1}}\ {\isacharminus}\ i{\isacharparenright}{\isacharparenright}\isanewline
-\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}i\ N{\isachardot}\ {\isasymnot}\ N\ {\isacharless}\ i\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isacharparenleft}Suc\ i{\isacharcomma}\ N{\isacharparenright}{\isacharcomma}\ i{\isacharcomma}\ N{\isacharparenright}\ {\isasymin}\ measure\ {\isacharparenleft}{\isasymlambda}{\isacharparenleft}i{\isacharcomma}\ N{\isacharparenright}{\isachardot}\ N\ {\isacharplus}\ {\isadigit{1}}\ {\isacharminus}\ i{\isacharparenright}%
-\end{isabelle}
-
- These goals are all solved by \isa{auto}:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-Let us complicate the function a little, by adding some more
- recursive calls:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ foo\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}foo\ i\ N\ {\isacharequal}\ {\isacharparenleft}if\ i\ {\isachargreater}\ N\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ then\ {\isacharparenleft}if\ N\ {\isacharequal}\ {\isadigit{0}}\ then\ {\isadigit{0}}\ else\ foo\ {\isadigit{0}}\ {\isacharparenleft}N\ {\isacharminus}\ {\isadigit{1}}{\isacharparenright}{\isacharparenright}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ else\ i\ {\isacharplus}\ foo\ {\isacharparenleft}Suc\ i{\isacharparenright}\ N{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ pat{\isacharunderscore}completeness\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-When \isa{i} has reached \isa{N}, it starts at zero again
- and \isa{N} is decremented.
- This corresponds to a nested
- loop where one index counts up and the other down. Termination can
- be proved using a lexicographic combination of two measures, namely
- the value of \isa{N} and the above difference. The \isa{measures} combinator generalizes \isa{measure} by taking a
- list of measure functions.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{termination}\isamarkupfalse%
-\ \isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}relation\ {\isachardoublequoteopen}measures\ {\isacharbrackleft}{\isasymlambda}{\isacharparenleft}i{\isacharcomma}\ N{\isacharparenright}{\isachardot}\ N{\isacharcomma}\ {\isasymlambda}{\isacharparenleft}i{\isacharcomma}N{\isacharparenright}{\isachardot}\ N\ {\isacharplus}\ {\isadigit{1}}\ {\isacharminus}\ i{\isacharbrackright}{\isachardoublequoteclose}{\isacharparenright}\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsubsection{How \isa{lexicographic{\isacharunderscore}order} works%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-To see how the automatic termination proofs work, let's look at an
- example where it fails\footnote{For a detailed discussion of the
- termination prover, see \cite{bulwahnKN07}}:
-
-\end{isamarkuptext}
-\cmd{fun} \isa{fails\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}nat\ {\isasymRightarrow}\ nat\ list\ {\isasymRightarrow}\ nat{\isachardoublequote}}\\%
-\cmd{where}\\%
-\hspace*{2ex}\isa{{\isachardoublequote}fails\ a\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharequal}\ a{\isachardoublequote}}\\%
-|\hspace*{1.5ex}\isa{{\isachardoublequote}fails\ a\ {\isacharparenleft}x{\isacharhash}xs{\isacharparenright}\ {\isacharequal}\ fails\ {\isacharparenleft}x\ {\isacharplus}\ a{\isacharparenright}\ {\isacharparenleft}x{\isacharhash}xs{\isacharparenright}{\isachardoublequote}}\\
-\begin{isamarkuptext}
-
-\noindent Isabelle responds with the following error:
-
-\begin{isabelle}
-*** Unfinished subgoals:\newline
-*** (a, 1, <):\newline
-*** \ 1.~\isa{{\isasymAnd}x{\isachardot}\ x\ {\isacharequal}\ {\isadigit{0}}}\newline
-*** (a, 1, <=):\newline
-*** \ 1.~False\newline
-*** (a, 2, <):\newline
-*** \ 1.~False\newline
-*** Calls:\newline
-*** a) \isa{{\isacharparenleft}a{\isacharcomma}\ x\ {\isacharhash}\ xs{\isacharparenright}\ {\isacharminus}{\isacharminus}{\isachargreater}{\isachargreater}\ {\isacharparenleft}x\ {\isacharplus}\ a{\isacharcomma}\ x\ {\isacharhash}\ xs{\isacharparenright}}\newline
-*** Measures:\newline
-*** 1) \isa{{\isasymlambda}x{\isachardot}\ size\ {\isacharparenleft}fst\ x{\isacharparenright}}\newline
-*** 2) \isa{{\isasymlambda}x{\isachardot}\ size\ {\isacharparenleft}snd\ x{\isacharparenright}}\newline
-*** Result matrix:\newline
-*** \ \ \ \ 1\ \ 2 \newline
-*** a: ? <= \newline
-*** Could not find lexicographic termination order.\newline
-*** At command "fun".\newline
-\end{isabelle}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The key to this error message is the matrix at the bottom. The rows
- of that matrix correspond to the different recursive calls (In our
- case, there is just one). The columns are the function's arguments
- (expressed through different measure functions, which map the
- argument tuple to a natural number).
-
- The contents of the matrix summarize what is known about argument
- descents: The second argument has a weak descent (\isa{{\isacharless}{\isacharequal}}) at the
- recursive call, and for the first argument nothing could be proved,
- which is expressed by \isa{{\isacharquery}}. In general, there are the values
- \isa{{\isacharless}}, \isa{{\isacharless}{\isacharequal}} and \isa{{\isacharquery}}.
-
- For the failed proof attempts, the unfinished subgoals are also
- printed. Looking at these will often point to a missing lemma.
-
-% As a more real example, here is quicksort:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{Mutual Recursion%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-If two or more functions call one another mutually, they have to be defined
- in one step. Here are \isa{even} and \isa{odd}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ even\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{and}\ odd\ \ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}even\ {\isadigit{0}}\ {\isacharequal}\ True{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}odd\ {\isadigit{0}}\ {\isacharequal}\ False{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}even\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ odd\ n{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}odd\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ even\ n{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ pat{\isacharunderscore}completeness\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-To eliminate the mutual dependencies, Isabelle internally
- creates a single function operating on the sum
- type \isa{nat\ {\isacharplus}\ nat}. Then, \isa{even} and \isa{odd} are
- defined as projections. Consequently, termination has to be proved
- simultaneously for both functions, by specifying a measure on the
- sum type:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{termination}\isamarkupfalse%
-\ \isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}relation\ {\isachardoublequoteopen}measure\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ case\ x\ of\ Inl\ n\ {\isasymRightarrow}\ n\ {\isacharbar}\ Inr\ n\ {\isasymRightarrow}\ n{\isacharparenright}{\isachardoublequoteclose}{\isacharparenright}\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-We could also have used \isa{lexicographic{\isacharunderscore}order}, which
- supports mutual recursive termination proofs to a certain extent.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Induction for mutual recursion%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-When functions are mutually recursive, proving properties about them
- generally requires simultaneous induction. The induction rule \isa{even{\isacharunderscore}odd{\isachardot}induct}
- generated from the above definition reflects this.
-
- Let us prove something about \isa{even} and \isa{odd}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ even{\isacharunderscore}odd{\isacharunderscore}mod{\isadigit{2}}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}even\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}odd\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{1}}{\isacharparenright}{\isachardoublequoteclose}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-%
-\begin{isamarkuptxt}%
-We apply simultaneous induction, specifying the induction variable
- for both goals, separated by \cmd{and}:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}induct\ n\ \isakeyword{and}\ n\ rule{\isacharcolon}\ even{\isacharunderscore}odd{\isachardot}induct{\isacharparenright}%
-\begin{isamarkuptxt}%
-We get four subgoals, which correspond to the clauses in the
- definition of \isa{even} and \isa{odd}:
- \begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ even\ {\isadigit{0}}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\isanewline
-\ {\isadigit{2}}{\isachardot}\ odd\ {\isadigit{0}}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{1}}{\isacharparenright}\isanewline
-\ {\isadigit{3}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ odd\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{1}}{\isacharparenright}\ {\isasymLongrightarrow}\ even\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Suc\ n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\isanewline
-\ {\isadigit{4}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ even\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\ {\isasymLongrightarrow}\ odd\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Suc\ n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{1}}{\isacharparenright}%
-\end{isabelle}
- Simplification solves the first two goals, leaving us with two
- statements about the \isa{mod} operation to prove:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ simp{\isacharunderscore}all%
-\begin{isamarkuptxt}%
-\begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ odd\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ Suc\ {\isadigit{0}}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ Suc\ {\isadigit{0}}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Suc\ n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\isanewline
-\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ even\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Suc\ n\ mod\ {\isadigit{2}}\ {\isacharequal}\ Suc\ {\isadigit{0}}{\isacharparenright}%
-\end{isabelle}
-
- \noindent These can be handled by Isabelle's arithmetic decision procedures.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ arith\isanewline
-\isacommand{apply}\isamarkupfalse%
-\ arith\isanewline
-\isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-In proofs like this, the simultaneous induction is really essential:
- Even if we are just interested in one of the results, the other
- one is necessary to strengthen the induction hypothesis. If we leave
- out the statement about \isa{odd} and just write \isa{True} instead,
- the same proof fails:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ failed{\isacharunderscore}attempt{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}even\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ {\isachardoublequoteopen}True{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}induct\ n\ rule{\isacharcolon}\ even{\isacharunderscore}odd{\isachardot}induct{\isacharparenright}%
-\begin{isamarkuptxt}%
-\noindent Now the third subgoal is a dead end, since we have no
- useful induction hypothesis available:
-
- \begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ even\ {\isadigit{0}}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\isanewline
-\ {\isadigit{2}}{\isachardot}\ True\isanewline
-\ {\isadigit{3}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ True\ {\isasymLongrightarrow}\ even\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Suc\ n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\isanewline
-\ {\isadigit{4}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ even\ n\ {\isacharequal}\ {\isacharparenleft}n\ mod\ {\isadigit{2}}\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}\ {\isasymLongrightarrow}\ True%
-\end{isabelle}%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{oops}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsection{General pattern matching%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\label{genpats}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Avoiding automatic pattern splitting%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Up to now, we used pattern matching only on datatypes, and the
- patterns were always disjoint and complete, and if they weren't,
- they were made disjoint automatically like in the definition of
- \isa{sep} in \S\ref{patmatch}.
-
- This automatic splitting can significantly increase the number of
- equations involved, and this is not always desirable. The following
- example shows the problem:
-
- Suppose we are modeling incomplete knowledge about the world by a
- three-valued datatype, which has values \isa{T}, \isa{F}
- and \isa{X} for true, false and uncertain propositions, respectively.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{datatype}\isamarkupfalse%
-\ P{\isadigit{3}}\ {\isacharequal}\ T\ {\isacharbar}\ F\ {\isacharbar}\ X%
-\begin{isamarkuptext}%
-\noindent Then the conjunction of such values can be defined as follows:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{fun}\isamarkupfalse%
-\ And\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}P{\isadigit{3}}\ {\isasymRightarrow}\ P{\isadigit{3}}\ {\isasymRightarrow}\ P{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}And\ T\ p\ {\isacharequal}\ p{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}And\ p\ T\ {\isacharequal}\ p{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}And\ p\ F\ {\isacharequal}\ F{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}And\ F\ p\ {\isacharequal}\ F{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}And\ X\ X\ {\isacharequal}\ X{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-This definition is useful, because the equations can directly be used
- as simplification rules. But the patterns overlap: For example,
- the expression \isa{And\ T\ T} is matched by both the first and
- the second equation. By default, Isabelle makes the patterns disjoint by
- splitting them up, producing instances:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{thm}\isamarkupfalse%
-\ And{\isachardot}simps%
-\begin{isamarkuptext}%
-\isa{And\ T\ {\isacharquery}p\ {\isacharequal}\ {\isacharquery}p\isasep\isanewline%
-And\ F\ T\ {\isacharequal}\ F\isasep\isanewline%
-And\ X\ T\ {\isacharequal}\ X\isasep\isanewline%
-And\ F\ F\ {\isacharequal}\ F\isasep\isanewline%
-And\ X\ F\ {\isacharequal}\ F\isasep\isanewline%
-And\ F\ X\ {\isacharequal}\ F\isasep\isanewline%
-And\ X\ X\ {\isacharequal}\ X}
-
- \vspace*{1em}
- \noindent There are several problems with this:
-
- \begin{enumerate}
- \item If the datatype has many constructors, there can be an
- explosion of equations. For \isa{And}, we get seven instead of
- five equations, which can be tolerated, but this is just a small
- example.
-
- \item Since splitting makes the equations \qt{less general}, they
- do not always match in rewriting. While the term \isa{And\ x\ F}
- can be simplified to \isa{F} with the original equations, a
- (manual) case split on \isa{x} is now necessary.
-
- \item The splitting also concerns the induction rule \isa{And{\isachardot}induct}. Instead of five premises it now has seven, which
- means that our induction proofs will have more cases.
-
- \item In general, it increases clarity if we get the same definition
- back which we put in.
- \end{enumerate}
-
- If we do not want the automatic splitting, we can switch it off by
- leaving out the \cmd{sequential} option. However, we will have to
- prove that our pattern matching is consistent\footnote{This prevents
- us from defining something like \isa{f\ x\ {\isacharequal}\ True} and \isa{f\ x\ {\isacharequal}\ False} simultaneously.}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ And{\isadigit{2}}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}P{\isadigit{3}}\ {\isasymRightarrow}\ P{\isadigit{3}}\ {\isasymRightarrow}\ P{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}And{\isadigit{2}}\ T\ p\ {\isacharequal}\ p{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}And{\isadigit{2}}\ p\ T\ {\isacharequal}\ p{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}And{\isadigit{2}}\ p\ F\ {\isacharequal}\ F{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}And{\isadigit{2}}\ F\ p\ {\isacharequal}\ F{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}And{\isadigit{2}}\ X\ X\ {\isacharequal}\ X{\isachardoublequoteclose}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-%
-\begin{isamarkuptxt}%
-\noindent Now let's look at the proof obligations generated by a
- function definition. In this case, they are:
-
- \begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}P\ x{\isachardot}\ {\isasymlbrakk}{\isasymAnd}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}T{\isacharcomma}\ p{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharsemicolon}\ {\isasymAnd}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}p{\isacharcomma}\ T{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharsemicolon}\ {\isasymAnd}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}p{\isacharcomma}\ F{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharsemicolon}\isanewline
-\isaindent{\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}P\ x{\isachardot}\ \ }{\isasymAnd}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}F{\isacharcomma}\ p{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isacharsemicolon}\ x\ {\isacharequal}\ {\isacharparenleft}X{\isacharcomma}\ X{\isacharparenright}\ {\isasymLongrightarrow}\ P{\isasymrbrakk}\isanewline
-\isaindent{\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}P\ x{\isachardot}\ }{\isasymLongrightarrow}\ P\isanewline
-\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}p\ pa{\isachardot}\ {\isacharparenleft}T{\isacharcomma}\ p{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}T{\isacharcomma}\ pa{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ pa\isanewline
-\ {\isadigit{3}}{\isachardot}\ {\isasymAnd}p\ pa{\isachardot}\ {\isacharparenleft}T{\isacharcomma}\ p{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}pa{\isacharcomma}\ T{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ pa\isanewline
-\ {\isadigit{4}}{\isachardot}\ {\isasymAnd}p\ pa{\isachardot}\ {\isacharparenleft}T{\isacharcomma}\ p{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}pa{\isacharcomma}\ F{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ F\isanewline
-\ {\isadigit{5}}{\isachardot}\ {\isasymAnd}p\ pa{\isachardot}\ {\isacharparenleft}T{\isacharcomma}\ p{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}F{\isacharcomma}\ pa{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ F\isanewline
-\ {\isadigit{6}}{\isachardot}\ {\isasymAnd}p{\isachardot}\ {\isacharparenleft}T{\isacharcomma}\ p{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}X{\isacharcomma}\ X{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ X\isanewline
-\ {\isadigit{7}}{\isachardot}\ {\isasymAnd}p\ pa{\isachardot}\ {\isacharparenleft}p{\isacharcomma}\ T{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}pa{\isacharcomma}\ T{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ pa\isanewline
-\ {\isadigit{8}}{\isachardot}\ {\isasymAnd}p\ pa{\isachardot}\ {\isacharparenleft}p{\isacharcomma}\ T{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}pa{\isacharcomma}\ F{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ F\isanewline
-\ {\isadigit{9}}{\isachardot}\ {\isasymAnd}p\ pa{\isachardot}\ {\isacharparenleft}p{\isacharcomma}\ T{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}F{\isacharcomma}\ pa{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ F\isanewline
-\ {\isadigit{1}}{\isadigit{0}}{\isachardot}\ {\isasymAnd}p{\isachardot}\ {\isacharparenleft}p{\isacharcomma}\ T{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}X{\isacharcomma}\ X{\isacharparenright}\ {\isasymLongrightarrow}\ p\ {\isacharequal}\ X%
-\end{isabelle}\vspace{-1.2em}\hspace{3cm}\vdots\vspace{1.2em}
-
- The first subgoal expresses the completeness of the patterns. It has
- the form of an elimination rule and states that every \isa{x} of
- the function's input type must match at least one of the patterns\footnote{Completeness could
- be equivalently stated as a disjunction of existential statements:
-\isa{{\isacharparenleft}{\isasymexists}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}T{\isacharcomma}\ p{\isacharparenright}{\isacharparenright}\ {\isasymor}\ {\isacharparenleft}{\isasymexists}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}p{\isacharcomma}\ T{\isacharparenright}{\isacharparenright}\ {\isasymor}\ {\isacharparenleft}{\isasymexists}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}p{\isacharcomma}\ F{\isacharparenright}{\isacharparenright}\ {\isasymor}\ {\isacharparenleft}{\isasymexists}p{\isachardot}\ x\ {\isacharequal}\ {\isacharparenleft}F{\isacharcomma}\ p{\isacharparenright}{\isacharparenright}\ {\isasymor}\ x\ {\isacharequal}\ {\isacharparenleft}X{\isacharcomma}\ X{\isacharparenright}}, and you can use the method \isa{atomize{\isacharunderscore}elim} to get that form instead.}. If the patterns just involve
- datatypes, we can solve it with the \isa{pat{\isacharunderscore}completeness}
- method:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ pat{\isacharunderscore}completeness%
-\begin{isamarkuptxt}%
-The remaining subgoals express \emph{pattern compatibility}. We do
- allow that an input value matches multiple patterns, but in this
- case, the result (i.e.~the right hand sides of the equations) must
- also be equal. For each pair of two patterns, there is one such
- subgoal. Usually this needs injectivity of the constructors, which
- is used automatically by \isa{auto}.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsubsection{Non-constructor patterns%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Most of Isabelle's basic types take the form of inductive datatypes,
- and usually pattern matching works on the constructors of such types.
- However, this need not be always the case, and the \cmd{function}
- command handles other kind of patterns, too.
-
- One well-known instance of non-constructor patterns are
- so-called \emph{$n+k$-patterns}, which are a little controversial in
- the functional programming world. Here is the initial fibonacci
- example with $n+k$-patterns:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ fib{\isadigit{2}}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}fib{\isadigit{2}}\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}fib{\isadigit{2}}\ {\isadigit{1}}\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}fib{\isadigit{2}}\ {\isacharparenleft}n\ {\isacharplus}\ {\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ fib{\isadigit{2}}\ n\ {\isacharplus}\ fib{\isadigit{2}}\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimML
-%
-\endisadelimML
-%
-\isatagML
-%
-\endisatagML
-{\isafoldML}%
-%
-\isadelimML
-%
-\endisadelimML
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-%
-\begin{isamarkuptxt}%
-This kind of matching is again justified by the proof of pattern
- completeness and compatibility.
- The proof obligation for pattern completeness states that every natural number is
- either \isa{{\isadigit{0}}}, \isa{{\isadigit{1}}} or \isa{n\ {\isacharplus}\ {\isadigit{2}}}:
-
- \begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}P\ x{\isachardot}\ {\isasymlbrakk}x\ {\isacharequal}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ P{\isacharsemicolon}\ x\ {\isacharequal}\ {\isadigit{1}}\ {\isasymLongrightarrow}\ P{\isacharsemicolon}\ {\isasymAnd}n{\isachardot}\ x\ {\isacharequal}\ n\ {\isacharplus}\ {\isadigit{2}}\ {\isasymLongrightarrow}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\isanewline
-\ {\isadigit{2}}{\isachardot}\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ {\isadigit{1}}\ {\isacharequal}\ {\isadigit{1}}\isanewline
-\ {\isadigit{3}}{\isachardot}\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{1}}\ {\isasymLongrightarrow}\ {\isadigit{1}}\ {\isacharequal}\ {\isadigit{1}}\isanewline
-\ {\isadigit{4}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ {\isadigit{0}}\ {\isacharequal}\ n\ {\isacharplus}\ {\isadigit{2}}\ {\isasymLongrightarrow}\ {\isadigit{1}}\ {\isacharequal}\ fib{\isadigit{2}}{\isacharunderscore}sumC\ n\ {\isacharplus}\ fib{\isadigit{2}}{\isacharunderscore}sumC\ {\isacharparenleft}Suc\ n{\isacharparenright}\isanewline
-\ {\isadigit{5}}{\isachardot}\ {\isadigit{1}}\ {\isacharequal}\ {\isadigit{1}}\ {\isasymLongrightarrow}\ {\isadigit{1}}\ {\isacharequal}\ {\isadigit{1}}\isanewline
-\ {\isadigit{6}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ {\isadigit{1}}\ {\isacharequal}\ n\ {\isacharplus}\ {\isadigit{2}}\ {\isasymLongrightarrow}\ {\isadigit{1}}\ {\isacharequal}\ fib{\isadigit{2}}{\isacharunderscore}sumC\ n\ {\isacharplus}\ fib{\isadigit{2}}{\isacharunderscore}sumC\ {\isacharparenleft}Suc\ n{\isacharparenright}\isanewline
-\ {\isadigit{7}}{\isachardot}\ {\isasymAnd}n\ na{\isachardot}\isanewline
-\isaindent{\ {\isadigit{7}}{\isachardot}\ \ \ \ }n\ {\isacharplus}\ {\isadigit{2}}\ {\isacharequal}\ na\ {\isacharplus}\ {\isadigit{2}}\ {\isasymLongrightarrow}\isanewline
-\isaindent{\ {\isadigit{7}}{\isachardot}\ \ \ \ }fib{\isadigit{2}}{\isacharunderscore}sumC\ n\ {\isacharplus}\ fib{\isadigit{2}}{\isacharunderscore}sumC\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ fib{\isadigit{2}}{\isacharunderscore}sumC\ na\ {\isacharplus}\ fib{\isadigit{2}}{\isacharunderscore}sumC\ {\isacharparenleft}Suc\ na{\isacharparenright}%
-\end{isabelle}
-
- This is an arithmetic triviality, but unfortunately the
- \isa{arith} method cannot handle this specific form of an
- elimination rule. However, we can use the method \isa{atomize{\isacharunderscore}elim} to do an ad-hoc conversion to a disjunction of
- existentials, which can then be solved by the arithmetic decision procedure.
- Pattern compatibility and termination are automatic as usual.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimML
-%
-\endisadelimML
-%
-\isatagML
-%
-\endisatagML
-{\isafoldML}%
-%
-\isadelimML
-%
-\endisadelimML
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ atomize{\isacharunderscore}elim\isanewline
-\isacommand{apply}\isamarkupfalse%
-\ arith\isanewline
-\isacommand{apply}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isacommand{termination}\isamarkupfalse%
-%
-\isadelimproof
-\ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ lexicographic{\isacharunderscore}order%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-We can stretch the notion of pattern matching even more. The
- following function is not a sensible functional program, but a
- perfectly valid mathematical definition:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ ev\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}ev\ {\isacharparenleft}{\isadigit{2}}\ {\isacharasterisk}\ n{\isacharparenright}\ {\isacharequal}\ True{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}ev\ {\isacharparenleft}{\isadigit{2}}\ {\isacharasterisk}\ n\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}\ {\isacharequal}\ False{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ atomize{\isacharunderscore}elim\isanewline
-\isacommand{by}\isamarkupfalse%
-\ arith{\isacharplus}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{termination}\isamarkupfalse%
-%
-\isadelimproof
-\ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}relation\ {\isachardoublequoteopen}{\isacharbraceleft}{\isacharbraceright}{\isachardoublequoteclose}{\isacharparenright}\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-This general notion of pattern matching gives you a certain freedom
- in writing down specifications. However, as always, such freedom should
- be used with care:
-
- If we leave the area of constructor
- patterns, we have effectively departed from the world of functional
- programming. This means that it is no longer possible to use the
- code generator, and expect it to generate ML code for our
- definitions. Also, such a specification might not work very well together with
- simplification. Your mileage may vary.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Conditional equations%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The function package also supports conditional equations, which are
- similar to guards in a language like Haskell. Here is Euclid's
- algorithm written with conditional patterns\footnote{Note that the
- patterns are also overlapping in the base case}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ gcd\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}gcd\ x\ {\isadigit{0}}\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}gcd\ {\isadigit{0}}\ y\ {\isacharequal}\ y{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}x\ {\isacharless}\ y\ {\isasymLongrightarrow}\ gcd\ {\isacharparenleft}Suc\ x{\isacharparenright}\ {\isacharparenleft}Suc\ y{\isacharparenright}\ {\isacharequal}\ gcd\ {\isacharparenleft}Suc\ x{\isacharparenright}\ {\isacharparenleft}y\ {\isacharminus}\ x{\isacharparenright}{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}{\isasymnot}\ x\ {\isacharless}\ y\ {\isasymLongrightarrow}\ gcd\ {\isacharparenleft}Suc\ x{\isacharparenright}\ {\isacharparenleft}Suc\ y{\isacharparenright}\ {\isacharequal}\ gcd\ {\isacharparenleft}x\ {\isacharminus}\ y{\isacharparenright}\ {\isacharparenleft}Suc\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}atomize{\isacharunderscore}elim{\isacharcomma}\ auto{\isacharcomma}\ arith{\isacharparenright}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{termination}\isamarkupfalse%
-%
-\isadelimproof
-\ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ lexicographic{\isacharunderscore}order%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-By now, you can probably guess what the proof obligations for the
- pattern completeness and compatibility look like.
-
- Again, functions with conditional patterns are not supported by the
- code generator.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Pattern matching on strings%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-As strings (as lists of characters) are normal datatypes, pattern
- matching on them is possible, but somewhat problematic. Consider the
- following definition:
-
-\end{isamarkuptext}
-\noindent\cmd{fun} \isa{check\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}string\ {\isasymRightarrow}\ bool{\isachardoublequote}}\\%
-\cmd{where}\\%
-\hspace*{2ex}\isa{{\isachardoublequote}check\ {\isacharparenleft}{\isacharprime}{\isacharprime}good{\isacharprime}{\isacharprime}{\isacharparenright}\ {\isacharequal}\ True{\isachardoublequote}}\\%
-\isa{{\isacharbar}\ {\isachardoublequote}check\ s\ {\isacharequal}\ False{\isachardoublequote}}
-\begin{isamarkuptext}
-
- \noindent An invocation of the above \cmd{fun} command does not
- terminate. What is the problem? Strings are lists of characters, and
- characters are a datatype with a lot of constructors. Splitting the
- catch-all pattern thus leads to an explosion of cases, which cannot
- be handled by Isabelle.
-
- There are two things we can do here. Either we write an explicit
- \isa{if} on the right hand side, or we can use conditional patterns:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ check\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}string\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}check\ {\isacharparenleft}{\isacharprime}{\isacharprime}good{\isacharprime}{\isacharprime}{\isacharparenright}\ {\isacharequal}\ True{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}s\ {\isasymnoteq}\ {\isacharprime}{\isacharprime}good{\isacharprime}{\isacharprime}\ {\isasymLongrightarrow}\ check\ s\ {\isacharequal}\ False{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsection{Partiality%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-In HOL, all functions are total. A function \isa{f} applied to
- \isa{x} always has the value \isa{f\ x}, and there is no notion
- of undefinedness.
- This is why we have to do termination
- proofs when defining functions: The proof justifies that the
- function can be defined by wellfounded recursion.
-
- However, the \cmd{function} package does support partiality to a
- certain extent. Let's look at the following function which looks
- for a zero of a given function f.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ findzero\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}nat\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}findzero\ f\ n\ {\isacharequal}\ {\isacharparenleft}if\ f\ n\ {\isacharequal}\ {\isadigit{0}}\ then\ n\ else\ findzero\ f\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ pat{\isacharunderscore}completeness\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\noindent Clearly, any attempt of a termination proof must fail. And without
- that, we do not get the usual rules \isa{findzero{\isachardot}simps} and
- \isa{findzero{\isachardot}induct}. So what was the definition good for at all?%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Domain predicates%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The trick is that Isabelle has not only defined the function \isa{findzero}, but also
- a predicate \isa{findzero{\isacharunderscore}dom} that characterizes the values where the function
- terminates: the \emph{domain} of the function. If we treat a
- partial function just as a total function with an additional domain
- predicate, we can derive simplification and
- induction rules as we do for total functions. They are guarded
- by domain conditions and are called \isa{psimps} and \isa{pinduct}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\noindent\begin{minipage}{0.79\textwidth}\begin{isabelle}%
-findzero{\isacharunderscore}dom\ {\isacharparenleft}{\isacharquery}f{\isacharcomma}\ {\isacharquery}n{\isacharparenright}\ {\isasymLongrightarrow}\isanewline
-findzero\ {\isacharquery}f\ {\isacharquery}n\ {\isacharequal}\ {\isacharparenleft}if\ {\isacharquery}f\ {\isacharquery}n\ {\isacharequal}\ {\isadigit{0}}\ then\ {\isacharquery}n\ else\ findzero\ {\isacharquery}f\ {\isacharparenleft}Suc\ {\isacharquery}n{\isacharparenright}{\isacharparenright}%
-\end{isabelle}\end{minipage}
- \hfill(\isa{findzero{\isachardot}psimps})
- \vspace{1em}
-
- \noindent\begin{minipage}{0.79\textwidth}\begin{isabelle}%
-{\isasymlbrakk}findzero{\isacharunderscore}dom\ {\isacharparenleft}{\isacharquery}a{\isadigit{0}}{\isachardot}{\isadigit{0}}{\isacharcomma}\ {\isacharquery}a{\isadigit{1}}{\isachardot}{\isadigit{0}}{\isacharparenright}{\isacharsemicolon}\isanewline
-\isaindent{\ }{\isasymAnd}f\ n{\isachardot}\ {\isasymlbrakk}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}{\isacharsemicolon}\ f\ n\ {\isasymnoteq}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ {\isacharquery}P\ f\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharquery}P\ f\ n{\isasymrbrakk}\isanewline
-{\isasymLongrightarrow}\ {\isacharquery}P\ {\isacharquery}a{\isadigit{0}}{\isachardot}{\isadigit{0}}\ {\isacharquery}a{\isadigit{1}}{\isachardot}{\isadigit{0}}%
-\end{isabelle}\end{minipage}
- \hfill(\isa{findzero{\isachardot}pinduct})%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Remember that all we
- are doing here is use some tricks to make a total function appear
- as if it was partial. We can still write the term \isa{findzero\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ {\isadigit{1}}{\isacharparenright}\ {\isadigit{0}}} and like any other term of type \isa{nat} it is equal
- to some natural number, although we might not be able to find out
- which one. The function is \emph{underdefined}.
-
- But it is defined enough to prove something interesting about it. We
- can prove that if \isa{findzero\ f\ n}
- terminates, it indeed returns a zero of \isa{f}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ findzero{\isacharunderscore}zero{\isacharcolon}\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}\ {\isasymLongrightarrow}\ f\ {\isacharparenleft}findzero\ f\ n{\isacharparenright}\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-%
-\begin{isamarkuptxt}%
-\noindent We apply induction as usual, but using the partial induction
- rule:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}induct\ f\ n\ rule{\isacharcolon}\ findzero{\isachardot}pinduct{\isacharparenright}%
-\begin{isamarkuptxt}%
-\noindent This gives the following subgoals:
-
- \begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}f\ n{\isachardot}\ {\isasymlbrakk}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}{\isacharsemicolon}\ f\ n\ {\isasymnoteq}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ f\ {\isacharparenleft}findzero\ f\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isadigit{0}}{\isasymrbrakk}\isanewline
-\isaindent{\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}f\ n{\isachardot}\ }{\isasymLongrightarrow}\ f\ {\isacharparenleft}findzero\ f\ n{\isacharparenright}\ {\isacharequal}\ {\isadigit{0}}%
-\end{isabelle}
-
- \noindent The hypothesis in our lemma was used to satisfy the first premise in
- the induction rule. However, we also get \isa{findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}} as a local assumption in the induction step. This
- allows to unfold \isa{findzero\ f\ n} using the \isa{psimps}
- rule, and the rest is trivial. Since the \isa{psimps} rules carry the
- \isa{{\isacharbrackleft}simp{\isacharbrackright}} attribute by default, we just need a single step:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-Proofs about partial functions are often not harder than for total
- functions. Fig.~\ref{findzero_isar} shows a slightly more
- complicated proof written in Isar. It is verbose enough to show how
- partiality comes into play: From the partial induction, we get an
- additional domain condition hypothesis. Observe how this condition
- is applied when calls to \isa{findzero} are unfolded.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{figure}
-\hrule\vspace{6pt}
-\begin{minipage}{0.8\textwidth}
-\isabellestyle{it}
-\isastyle\isamarkuptrue
-\isacommand{lemma}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymlbrakk}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}{\isacharsemicolon}\ x\ {\isasymin}\ {\isacharbraceleft}n\ {\isachardot}{\isachardot}{\isacharless}\ findzero\ f\ n{\isacharbraceright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ f\ x\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}induct\ rule{\isacharcolon}\ findzero{\isachardot}pinduct{\isacharparenright}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ f\ n\ \isacommand{assume}\isamarkupfalse%
-\ dom{\isacharcolon}\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \isakeyword{and}\ IH{\isacharcolon}\ {\isachardoublequoteopen}{\isasymlbrakk}f\ n\ {\isasymnoteq}\ {\isadigit{0}}{\isacharsemicolon}\ x\ {\isasymin}\ {\isacharbraceleft}Suc\ n\ {\isachardot}{\isachardot}{\isacharless}\ findzero\ f\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isacharbraceright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ f\ x\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \isakeyword{and}\ x{\isacharunderscore}range{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}n\ {\isachardot}{\isachardot}{\isacharless}\ findzero\ f\ n{\isacharbraceright}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}f\ n\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ \isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}f\ n\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ dom\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}findzero\ f\ n\ {\isacharequal}\ n{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ x{\isacharunderscore}range\ \isacommand{show}\isamarkupfalse%
-\ False\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ x{\isacharunderscore}range\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharequal}\ n\ {\isasymor}\ x\ {\isasymin}\ {\isacharbraceleft}Suc\ n\ {\isachardot}{\isachardot}{\isacharless}\ findzero\ f\ n{\isacharbraceright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}f\ x\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharequal}\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ {\isacharbackquoteopen}f\ n\ {\isasymnoteq}\ {\isadigit{0}}{\isacharbackquoteclose}\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}thesis\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}Suc\ n\ {\isachardot}{\isachardot}{\isacharless}\ findzero\ f\ n{\isacharbraceright}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ dom\ \isakeyword{and}\ {\isacharbackquoteopen}f\ n\ {\isasymnoteq}\ {\isadigit{0}}{\isacharbackquoteclose}\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}Suc\ n\ {\isachardot}{\isachardot}{\isacharless}\ findzero\ f\ {\isacharparenleft}Suc\ n{\isacharparenright}{\isacharbraceright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ IH\ \isakeyword{and}\ {\isacharbackquoteopen}f\ n\ {\isasymnoteq}\ {\isadigit{0}}{\isacharbackquoteclose}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}thesis\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupfalse\isabellestyle{tt}
-\end{minipage}\vspace{6pt}\hrule
-\caption{A proof about a partial function}\label{findzero_isar}
-\end{figure}
-%
-\isamarkupsubsection{Partial termination proofs%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Now that we have proved some interesting properties about our
- function, we should turn to the domain predicate and see if it is
- actually true for some values. Otherwise we would have just proved
- lemmas with \isa{False} as a premise.
-
- Essentially, we need some introduction rules for \isa{findzero{\isacharunderscore}dom}. The function package can prove such domain
- introduction rules automatically. But since they are not used very
- often (they are almost never needed if the function is total), this
- functionality is disabled by default for efficiency reasons. So we have to go
- back and ask for them explicitly by passing the \isa{{\isacharparenleft}domintros{\isacharparenright}} option to the function package:
-
-\vspace{1ex}
-\noindent\cmd{function} \isa{{\isacharparenleft}domintros{\isacharparenright}\ findzero\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}nat\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat{\isachardoublequote}}\\%
-\cmd{where}\isanewline%
-\ \ \ldots\\
-
- \noindent Now the package has proved an introduction rule for \isa{findzero{\isacharunderscore}dom}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{thm}\isamarkupfalse%
-\ findzero{\isachardot}domintros%
-\begin{isamarkuptext}%
-\begin{isabelle}%
-{\isacharparenleft}{\isadigit{0}}\ {\isacharless}\ {\isacharquery}f\ {\isacharquery}n\ {\isasymLongrightarrow}\ findzero{\isacharunderscore}dom\ {\isacharparenleft}{\isacharquery}f{\isacharcomma}\ Suc\ {\isacharquery}n{\isacharparenright}{\isacharparenright}\ {\isasymLongrightarrow}\ findzero{\isacharunderscore}dom\ {\isacharparenleft}{\isacharquery}f{\isacharcomma}\ {\isacharquery}n{\isacharparenright}%
-\end{isabelle}
-
- Domain introduction rules allow to show that a given value lies in the
- domain of a function, if the arguments of all recursive calls
- are in the domain as well. They allow to do a \qt{single step} in a
- termination proof. Usually, you want to combine them with a suitable
- induction principle.
-
- Since our function increases its argument at recursive calls, we
- need an induction principle which works \qt{backwards}. We will use
- \isa{inc{\isacharunderscore}induct}, which allows to do induction from a fixed number
- \qt{downwards}:
-
- \begin{center}\isa{{\isasymlbrakk}{\isacharquery}i\ {\isasymle}\ {\isacharquery}j{\isacharsemicolon}\ {\isacharquery}P\ {\isacharquery}j{\isacharsemicolon}\ {\isasymAnd}i{\isachardot}\ {\isasymlbrakk}i\ {\isacharless}\ {\isacharquery}j{\isacharsemicolon}\ {\isacharquery}P\ {\isacharparenleft}Suc\ i{\isacharparenright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharquery}P\ i{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharquery}P\ {\isacharquery}i}\hfill(\isa{inc{\isacharunderscore}induct})\end{center}
-
- Figure \ref{findzero_term} gives a detailed Isar proof of the fact
- that \isa{findzero} terminates if there is a zero which is greater
- or equal to \isa{n}. First we derive two useful rules which will
- solve the base case and the step case of the induction. The
- induction is then straightforward, except for the unusual induction
- principle.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{figure}
-\hrule\vspace{6pt}
-\begin{minipage}{0.8\textwidth}
-\isabellestyle{it}
-\isastyle\isamarkuptrue
-\isacommand{lemma}\isamarkupfalse%
-\ findzero{\isacharunderscore}termination{\isacharcolon}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}x\ {\isasymge}\ n{\isachardoublequoteclose}\ \isakeyword{and}\ {\isachardoublequoteopen}f\ x\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ base{\isacharcolon}\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ x{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}rule\ findzero{\isachardot}domintros{\isacharparenright}\ {\isacharparenleft}simp\ add{\isacharcolon}{\isacharbackquoteopen}f\ x\ {\isacharequal}\ {\isadigit{0}}{\isacharbackquoteclose}{\isacharparenright}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ step{\isacharcolon}\ {\isachardoublequoteopen}{\isasymAnd}i{\isachardot}\ findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ Suc\ i{\isacharparenright}\ \isanewline
-\ \ \ \ {\isasymLongrightarrow}\ findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ i{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}rule\ findzero{\isachardot}domintros{\isacharparenright}\ simp\isanewline
-\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isacharbackquoteopen}x\ {\isasymge}\ n{\isacharbackquoteclose}\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}induct\ rule{\isacharcolon}inc{\isacharunderscore}induct{\isacharparenright}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ x{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}rule\ base{\isacharparenright}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ i\ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ Suc\ i{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ i{\isacharparenright}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}rule\ step{\isacharparenright}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupfalse\isabellestyle{tt}
-\end{minipage}\vspace{6pt}\hrule
-\caption{Termination proof for \isa{findzero}}\label{findzero_term}
-\end{figure}
-%
-\begin{isamarkuptext}%
-Again, the proof given in Fig.~\ref{findzero_term} has a lot of
- detail in order to explain the principles. Using more automation, we
- can also have a short proof:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ findzero{\isacharunderscore}termination{\isacharunderscore}short{\isacharcolon}\isanewline
-\ \ \isakeyword{assumes}\ zero{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isachargreater}{\isacharequal}\ n{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}f\ x\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ zero\isanewline
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}induct\ rule{\isacharcolon}inc{\isacharunderscore}induct{\isacharparenright}\ {\isacharparenleft}auto\ intro{\isacharcolon}\ findzero{\isachardot}domintros{\isacharparenright}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\noindent It is simple to combine the partial correctness result with the
- termination lemma:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ findzero{\isacharunderscore}total{\isacharunderscore}correctness{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}f\ x\ {\isacharequal}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ f\ {\isacharparenleft}findzero\ f\ {\isadigit{0}}{\isacharparenright}\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}blast\ intro{\isacharcolon}\ findzero{\isacharunderscore}zero\ findzero{\isacharunderscore}termination{\isacharparenright}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsubsection{Definition of the domain predicate%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Sometimes it is useful to know what the definition of the domain
- predicate looks like. Actually, \isa{findzero{\isacharunderscore}dom} is just an
- abbreviation:
-
- \begin{isabelle}%
-findzero{\isacharunderscore}dom\ {\isasymequiv}\ accp\ findzero{\isacharunderscore}rel%
-\end{isabelle}
-
- The domain predicate is the \emph{accessible part} of a relation \isa{findzero{\isacharunderscore}rel}, which was also created internally by the function
- package. \isa{findzero{\isacharunderscore}rel} is just a normal
- inductive predicate, so we can inspect its definition by
- looking at the introduction rules \isa{findzero{\isacharunderscore}rel{\isachardot}intros}.
- In our case there is just a single rule:
-
- \begin{isabelle}%
-{\isacharquery}f\ {\isacharquery}n\ {\isasymnoteq}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ findzero{\isacharunderscore}rel\ {\isacharparenleft}{\isacharquery}f{\isacharcomma}\ Suc\ {\isacharquery}n{\isacharparenright}\ {\isacharparenleft}{\isacharquery}f{\isacharcomma}\ {\isacharquery}n{\isacharparenright}%
-\end{isabelle}
-
- The predicate \isa{findzero{\isacharunderscore}rel}
- describes the \emph{recursion relation} of the function
- definition. The recursion relation is a binary relation on
- the arguments of the function that relates each argument to its
- recursive calls. In general, there is one introduction rule for each
- recursive call.
-
- The predicate \isa{findzero{\isacharunderscore}dom} is the accessible part of
- that relation. An argument belongs to the accessible part, if it can
- be reached in a finite number of steps (cf.~its definition in \isa{Wellfounded{\isachardot}thy}).
-
- Since the domain predicate is just an abbreviation, you can use
- lemmas for \isa{accp} and \isa{findzero{\isacharunderscore}rel} directly. Some
- lemmas which are occasionally useful are \isa{accpI}, \isa{accp{\isacharunderscore}downward}, and of course the introduction and elimination rules
- for the recursion relation \isa{findzero{\isachardot}intros} and \isa{findzero{\isachardot}cases}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{A Useful Special Case: Tail recursion%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The domain predicate is our trick that allows us to model partiality
- in a world of total functions. The downside of this is that we have
- to carry it around all the time. The termination proof above allowed
- us to replace the abstract \isa{findzero{\isacharunderscore}dom\ {\isacharparenleft}f{\isacharcomma}\ n{\isacharparenright}} by the more
- concrete \isa{n\ {\isasymle}\ x\ {\isasymand}\ f\ x\ {\isacharequal}\ {\isadigit{0}}}, but the condition is still
- there and can only be discharged for special cases.
- In particular, the domain predicate guards the unfolding of our
- function, since it is there as a condition in the \isa{psimp}
- rules.
-
- Now there is an important special case: We can actually get rid
- of the condition in the simplification rules, \emph{if the function
- is tail-recursive}. The reason is that for all tail-recursive
- equations there is a total function satisfying them, even if they
- are non-terminating.
-
-% A function is tail recursive, if each call to the function is either
-% equal
-%
-% So the outer form of the
-%
-%if it can be written in the following
-% form:
-% {term[display] "f x = (if COND x then BASE x else f (LOOP x))"}
-
-
- The function package internally does the right construction and can
- derive the unconditional simp rules, if we ask it to do so. Luckily,
- our \isa{findzero} function is tail-recursive, so we can just go
- back and add another option to the \cmd{function} command:
-
-\vspace{1ex}
-\noindent\cmd{function} \isa{{\isacharparenleft}domintros{\isacharcomma}\ tailrec{\isacharparenright}\ findzero\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}nat\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ nat{\isachardoublequote}}\\%
-\cmd{where}\isanewline%
-\ \ \ldots\\%
-
-
- \noindent Now, we actually get unconditional simplification rules, even
- though the function is partial:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{thm}\isamarkupfalse%
-\ findzero{\isachardot}simps%
-\begin{isamarkuptext}%
-\begin{isabelle}%
-findzero\ {\isacharquery}f\ {\isacharquery}n\ {\isacharequal}\ {\isacharparenleft}if\ {\isacharquery}f\ {\isacharquery}n\ {\isacharequal}\ {\isadigit{0}}\ then\ {\isacharquery}n\ else\ findzero\ {\isacharquery}f\ {\isacharparenleft}Suc\ {\isacharquery}n{\isacharparenright}{\isacharparenright}%
-\end{isabelle}
-
- \noindent Of course these would make the simplifier loop, so we better remove
- them from the simpset:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{declare}\isamarkupfalse%
-\ findzero{\isachardot}simps{\isacharbrackleft}simp\ del{\isacharbrackright}%
-\begin{isamarkuptext}%
-Getting rid of the domain conditions in the simplification rules is
- not only useful because it simplifies proofs. It is also required in
- order to use Isabelle's code generator to generate ML code
- from a function definition.
- Since the code generator only works with equations, it cannot be
- used with \isa{psimp} rules. Thus, in order to generate code for
- partial functions, they must be defined as a tail recursion.
- Luckily, many functions have a relatively natural tail recursive
- definition.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{Nested recursion%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Recursive calls which are nested in one another frequently cause
- complications, since their termination proof can depend on a partial
- correctness property of the function itself.
-
- As a small example, we define the \qt{nested zero} function:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ nz\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}nz\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}nz\ {\isacharparenleft}Suc\ n{\isacharparenright}\ {\isacharequal}\ nz\ {\isacharparenleft}nz\ n{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ pat{\isacharunderscore}completeness\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-If we attempt to prove termination using the identity measure on
- naturals, this fails:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{termination}\isamarkupfalse%
-\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ {\isacharparenleft}relation\ {\isachardoublequoteopen}measure\ {\isacharparenleft}{\isasymlambda}n{\isachardot}\ n{\isacharparenright}{\isachardoublequoteclose}{\isacharparenright}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ auto%
-\begin{isamarkuptxt}%
-We get stuck with the subgoal
-
- \begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}n{\isachardot}\ nz{\isacharunderscore}dom\ n\ {\isasymLongrightarrow}\ nz\ n\ {\isacharless}\ Suc\ n%
-\end{isabelle}
-
- Of course this statement is true, since we know that \isa{nz} is
- the zero function. And in fact we have no problem proving this
- property by induction.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isacommand{lemma}\isamarkupfalse%
-\ nz{\isacharunderscore}is{\isacharunderscore}zero{\isacharcolon}\ {\isachardoublequoteopen}nz{\isacharunderscore}dom\ n\ {\isasymLongrightarrow}\ nz\ n\ {\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}induct\ rule{\isacharcolon}nz{\isachardot}pinduct{\isacharparenright}\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-We formulate this as a partial correctness lemma with the condition
- \isa{nz{\isacharunderscore}dom\ n}. This allows us to prove it with the \isa{pinduct} rule before we have proved termination. With this lemma,
- the termination proof works as expected:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{termination}\isamarkupfalse%
-\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}relation\ {\isachardoublequoteopen}measure\ {\isacharparenleft}{\isasymlambda}n{\isachardot}\ n{\isacharparenright}{\isachardoublequoteclose}{\isacharparenright}\ {\isacharparenleft}auto\ simp{\isacharcolon}\ nz{\isacharunderscore}is{\isacharunderscore}zero{\isacharparenright}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-As a general strategy, one should prove the statements needed for
- termination as a partial property first. Then they can be used to do
- the termination proof. This also works for less trivial
- examples. Figure \ref{f91} defines the 91-function, a well-known
- challenge problem due to John McCarthy, and proves its termination.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{figure}
-\hrule\vspace{6pt}
-\begin{minipage}{0.8\textwidth}
-\isabellestyle{it}
-\isastyle\isamarkuptrue
-\isacommand{function}\isamarkupfalse%
-\ f{\isadigit{9}}{\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}f{\isadigit{9}}{\isadigit{1}}\ n\ {\isacharequal}\ {\isacharparenleft}if\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isacharless}\ n\ then\ n\ {\isacharminus}\ {\isadigit{1}}{\isadigit{0}}\ else\ f{\isadigit{9}}{\isadigit{1}}\ {\isacharparenleft}f{\isadigit{9}}{\isadigit{1}}\ {\isacharparenleft}n\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ pat{\isacharunderscore}completeness\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ f{\isadigit{9}}{\isadigit{1}}{\isacharunderscore}estimate{\isacharcolon}\ \isanewline
-\ \ \isakeyword{assumes}\ trm{\isacharcolon}\ {\isachardoublequoteopen}f{\isadigit{9}}{\isadigit{1}}{\isacharunderscore}dom\ n{\isachardoublequoteclose}\ \isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}n\ {\isacharless}\ f{\isadigit{9}}{\isadigit{1}}\ n\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ trm\ \isacommand{by}\isamarkupfalse%
-\ induct\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{termination}\isamarkupfalse%
-\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}R\ {\isacharequal}\ {\isachardoublequoteopen}measure\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{1}}\ {\isacharminus}\ x{\isacharparenright}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}wf\ {\isacharquery}R{\isachardoublequoteclose}\ \isacommand{{\isachardot}{\isachardot}}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ n\ {\isacharcolon}{\isacharcolon}\ nat\ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnot}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isacharless}\ n{\isachardoublequoteclose}\ %
-\isamarkupcmt{Assumptions for both calls%
-}
-\isanewline
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}n\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}{\isacharcomma}\ n{\isacharparenright}\ {\isasymin}\ {\isacharquery}R{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ simp\ %
-\isamarkupcmt{Inner call%
-}
-\isanewline
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ inner{\isacharunderscore}trm{\isacharcolon}\ {\isachardoublequoteopen}f{\isadigit{9}}{\isadigit{1}}{\isacharunderscore}dom\ {\isacharparenleft}n\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}{\isacharparenright}{\isachardoublequoteclose}\ %
-\isamarkupcmt{Outer call%
-}
-\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ f{\isadigit{9}}{\isadigit{1}}{\isacharunderscore}estimate\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}n\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}\ {\isacharless}\ f{\isadigit{9}}{\isadigit{1}}\ {\isacharparenleft}n\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}{\isacharparenright}\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}{\isachardoublequoteclose}\ \isacommand{{\isachardot}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ {\isacharbackquoteopen}{\isasymnot}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isacharless}\ n{\isacharbackquoteclose}\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}f{\isadigit{9}}{\isadigit{1}}\ {\isacharparenleft}n\ {\isacharplus}\ {\isadigit{1}}{\isadigit{1}}{\isacharparenright}{\isacharcomma}\ n{\isacharparenright}\ {\isasymin}\ {\isacharquery}R{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupfalse\isabellestyle{tt}
-\end{minipage}
-\vspace{6pt}\hrule
-\caption{McCarthy's 91-function}\label{f91}
-\end{figure}
-%
-\isamarkupsection{Higher-Order Recursion%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Higher-order recursion occurs when recursive calls
- are passed as arguments to higher-order combinators such as \isa{map}, \isa{filter} etc.
- As an example, imagine a datatype of n-ary trees:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{datatype}\isamarkupfalse%
-\ {\isacharprime}a\ tree\ {\isacharequal}\ \isanewline
-\ \ Leaf\ {\isacharprime}a\ \isanewline
-{\isacharbar}\ Branch\ {\isachardoublequoteopen}{\isacharprime}a\ tree\ list{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\noindent We can define a function which swaps the left and right subtrees recursively, using the
- list functions \isa{rev} and \isa{map}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{fun}\isamarkupfalse%
-\ mirror\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ tree\ {\isasymRightarrow}\ {\isacharprime}a\ tree{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}mirror\ {\isacharparenleft}Leaf\ n{\isacharparenright}\ {\isacharequal}\ Leaf\ n{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}mirror\ {\isacharparenleft}Branch\ l{\isacharparenright}\ {\isacharequal}\ Branch\ {\isacharparenleft}rev\ {\isacharparenleft}map\ mirror\ l{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-Although the definition is accepted without problems, let us look at the termination proof:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{termination}\isamarkupfalse%
-%
-\isadelimproof
-\ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-%
-\begin{isamarkuptxt}%
-As usual, we have to give a wellfounded relation, such that the
- arguments of the recursive calls get smaller. But what exactly are
- the arguments of the recursive calls when mirror is given as an
- argument to \isa{map}? Isabelle gives us the
- subgoals
-
- \begin{isabelle}%
-\ {\isadigit{1}}{\isachardot}\ wf\ {\isacharquery}R\isanewline
-\ {\isadigit{2}}{\isachardot}\ {\isasymAnd}l\ x{\isachardot}\ x\ {\isasymin}\ set\ l\ {\isasymLongrightarrow}\ {\isacharparenleft}x{\isacharcomma}\ Branch\ l{\isacharparenright}\ {\isasymin}\ {\isacharquery}R%
-\end{isabelle}
-
- So the system seems to know that \isa{map} only
- applies the recursive call \isa{mirror} to elements
- of \isa{l}, which is essential for the termination proof.
-
- This knowledge about \isa{map} is encoded in so-called congruence rules,
- which are special theorems known to the \cmd{function} command. The
- rule for \isa{map} is
-
- \begin{isabelle}%
-{\isasymlbrakk}{\isacharquery}xs\ {\isacharequal}\ {\isacharquery}ys{\isacharsemicolon}\ {\isasymAnd}x{\isachardot}\ x\ {\isasymin}\ set\ {\isacharquery}ys\ {\isasymLongrightarrow}\ {\isacharquery}f\ x\ {\isacharequal}\ {\isacharquery}g\ x{\isasymrbrakk}\ {\isasymLongrightarrow}\ map\ {\isacharquery}f\ {\isacharquery}xs\ {\isacharequal}\ map\ {\isacharquery}g\ {\isacharquery}ys%
-\end{isabelle}
-
- You can read this in the following way: Two applications of \isa{map} are equal, if the list arguments are equal and the functions
- coincide on the elements of the list. This means that for the value
- \isa{map\ f\ l} we only have to know how \isa{f} behaves on
- the elements of \isa{l}.
-
- Usually, one such congruence rule is
- needed for each higher-order construct that is used when defining
- new functions. In fact, even basic functions like \isa{If} and \isa{Let} are handled by this mechanism. The congruence
- rule for \isa{If} states that the \isa{then} branch is only
- relevant if the condition is true, and the \isa{else} branch only if it
- is false:
-
- \begin{isabelle}%
-{\isasymlbrakk}{\isacharquery}b\ {\isacharequal}\ {\isacharquery}c{\isacharsemicolon}\ {\isacharquery}c\ {\isasymLongrightarrow}\ {\isacharquery}x\ {\isacharequal}\ {\isacharquery}u{\isacharsemicolon}\ {\isasymnot}\ {\isacharquery}c\ {\isasymLongrightarrow}\ {\isacharquery}y\ {\isacharequal}\ {\isacharquery}v{\isasymrbrakk}\isanewline
-{\isasymLongrightarrow}\ {\isacharparenleft}if\ {\isacharquery}b\ then\ {\isacharquery}x\ else\ {\isacharquery}y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}if\ {\isacharquery}c\ then\ {\isacharquery}u\ else\ {\isacharquery}v{\isacharparenright}%
-\end{isabelle}
-
- Congruence rules can be added to the
- function package by giving them the \isa{fundef{\isacharunderscore}cong} attribute.
-
- The constructs that are predefined in Isabelle, usually
- come with the respective congruence rules.
- But if you define your own higher-order functions, you may have to
- state and prove the required congruence rules yourself, if you want to use your
- functions in recursive definitions.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsubsection{Congruence Rules and Evaluation Order%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Higher order logic differs from functional programming languages in
- that it has no built-in notion of evaluation order. A program is
- just a set of equations, and it is not specified how they must be
- evaluated.
-
- However for the purpose of function definition, we must talk about
- evaluation order implicitly, when we reason about termination.
- Congruence rules express that a certain evaluation order is
- consistent with the logical definition.
-
- Consider the following function.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ f\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}f\ n\ {\isacharequal}\ {\isacharparenleft}n\ {\isacharequal}\ {\isadigit{0}}\ {\isasymor}\ f\ {\isacharparenleft}n\ {\isacharminus}\ {\isadigit{1}}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-For this definition, the termination proof fails. The default configuration
- specifies no congruence rule for disjunction. We have to add a
- congruence rule that specifies left-to-right evaluation order:
-
- \vspace{1ex}
- \noindent \isa{{\isasymlbrakk}{\isacharquery}P\ {\isacharequal}\ {\isacharquery}P{\isacharprime}{\isacharsemicolon}\ {\isasymnot}\ {\isacharquery}P{\isacharprime}\ {\isasymLongrightarrow}\ {\isacharquery}Q\ {\isacharequal}\ {\isacharquery}Q{\isacharprime}{\isasymrbrakk}\ {\isasymLongrightarrow}\ {\isacharparenleft}{\isacharquery}P\ {\isasymor}\ {\isacharquery}Q{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isacharquery}P{\isacharprime}\ {\isasymor}\ {\isacharquery}Q{\isacharprime}{\isacharparenright}}\hfill(\isa{disj{\isacharunderscore}cong})
- \vspace{1ex}
-
- Now the definition works without problems. Note how the termination
- proof depends on the extra condition that we get from the congruence
- rule.
-
- However, as evaluation is not a hard-wired concept, we
- could just turn everything around by declaring a different
- congruence rule. Then we can make the reverse definition:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ disj{\isacharunderscore}cong{\isadigit{2}}{\isacharbrackleft}fundef{\isacharunderscore}cong{\isacharbrackright}{\isacharcolon}\ \isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\isasymnot}\ Q{\isacharprime}\ {\isasymLongrightarrow}\ P\ {\isacharequal}\ P{\isacharprime}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}Q\ {\isacharequal}\ Q{\isacharprime}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isacharparenleft}P\ {\isasymor}\ Q{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}P{\isacharprime}\ {\isasymor}\ Q{\isacharprime}{\isacharparenright}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ f{\isacharprime}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}f{\isacharprime}\ n\ {\isacharequal}\ {\isacharparenleft}f{\isacharprime}\ {\isacharparenleft}n\ {\isacharminus}\ {\isadigit{1}}{\isacharparenright}\ {\isasymor}\ n\ {\isacharequal}\ {\isadigit{0}}{\isacharparenright}{\isachardoublequoteclose}%
-\begin{isamarkuptext}%
-\noindent These examples show that, in general, there is no \qt{best} set of
- congruence rules.
-
- However, such tweaking should rarely be necessary in
- practice, as most of the time, the default set of congruence rules
- works well.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\isanewline
-\end{isabellebody}%
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